Abstract of thesis entitled. Submitted by. Lei Chi-Un. for the degree of Doctor of Philosophy at the University of Hong Kong in January 2011

Transcription

1 Abstract of thesis entitled VLSI Macromodeling and Signal Integrity Analysis via Digital Signal Processing Techniques Submitted by Lei Chi-Un for the degree of Doctor of Philosophy at the University of Hong Kong in January 2011 With the increasing operation frequency and decreasing feature size of very-large-scale integration (VLSI) circuits, high-frequency effects, such as signal delay, crosstalk and simultaneous switching noise, have become a dominant factor limiting integrated circuit (IC) system performance. Accurate and efficient simulation is required during the IC design phase to capture the high-frequency behaviours of electronic systems. Linear macromodeling, in this context, refers to replacing a high-order system by a small-order linear model with similar input-output responses, for computationally efficient simulation and timecritical design. Macromodels can be generated by fitting tabulated data from measurement/simulation. There are a number of stringent modeling constraints in the non-linear computation for high-frequency and/or large scale systems, such as accuracy, computation complexity, manual intervention and numerical robustness. Due to the strict requirements of the modeling problem and developments of emerging technologies, there is no optimal algorithm so far, making macromodeling a challenging problem and a high value research topic. Furthermore, pre- and post-processing techniques and improvements from non-control-theoretic perspectives have been less explored. Recognizing the fundamental concept that sampled response is a discretized signal

2 sequence, this thesis explores the feasibility and benefits of applying digital signal processing (DSP) techniques to the macromodeling process of VLSI simulation. Several ideas from DSP have been proposed to facilitate this process: 1. Exploring the macromodeling process in signal/power integrity analyses in a system identification perspective, along three directions: data, algorithms and models, which builds up a substantial foundation for the future development of the macromodeling process; 2. Introducing a family of functionality-oriented extensions (numerically robust computation and new applications / approximation criteria / peripheral pre-processing support) to a widely adopted Vector Fitting macromodeling framework, based on the numerically robust discrete-time domain computation, which improves the accuracy and functionality of the existing macromodeling process; 3. Developing a new time-domain linear macromodeling algorithm through applying Walsh theorem and complementary signal techniques, which significantly simplifies the computation procedure by avoiding pole-sensitive and eigenvalue computations, and thus improves the fitting accuracy and speed; 4. Introducing a pre-processing technique, frequency warping, through re-distributing the spectral information of the response, which improves the numerical conditioning in the computation and facilitate the computation process. These techniques have been shown to significantly improve the robustness and efficiency of the macromodeling process, as demonstrated in their applications to industrial benchmark examples. The innovative application of DSP techniques has provided a new research paradigm for the advancement of the VLSI macromodeling process and circuit simulation. Abstract word count: 405

3 VLSI Macromodeling and Signal Integrity Analysis via Digital Signal Processing Techniques by LEI Chi-Un B.Eng. (EComE), The University of Hong Kong A thesis submitted in partial fulfilment of the requirements for the Degree of Doctor of Philosophy at The University of Hong Kong. January 2011

4 Declaration I declare that this thesis represents my own work, except where due acknowledgement is made, and that it has not been previously included in a thesis, dissertation or report submitted to this University or to any other institution for a degree, diploma or other qualifications. Signed LEI Chi-Un i

5 Acknowledgements First of all, I would like to thank Dr. N. Wong and Prof. T.S. Ng, my advisors and outstanding scientists, for their advices during my graduate study. I would like to extend my gratitude to all current and graduated members of the HKUEEE VLSI Design group. I would like to offer a special thank to Prof. H.K. Kwan at Winsdor University, Canada, for his support to my graduate study. I would also like to thank my parents and my brother for their love and encouragement, as well as my churchmates for their earnest prayers. Finally, I would like to thank God for always leading me. May God use me and glory be to Him. Trust in the LORD with all thine heart; and lean not unto thine own understanding. In all thy ways acknowledge him, and he shall direct thy paths. (Proverbs 3:5-6) Behold, I will do a new thing; now it shall spring forth; shall ye not know it? I will even make a way in the wilderness, and rivers in the desert. (Isaiah 43:19) Chi-Un LEI May 2010 ii

6 Contents Declaration i Acknowledgements ii Table of Contents iii List of Tables viii List of Figures ix List of Abbreviations xiv 1 Introduction IC design: In the Signal-Integrity-Aware Era Signal Integrity: The Physical Interconnect Affection of the Electrical Performance in Circuits Macromodeling: System Identification Problem in Signal/Power Integrity Digital Signal Processing: Processing Sampled and Discretized Signals Research Motivations and Dissertation Outline Macromodeling Process Developments in the Vector Fitting Based Framework and Discrete-Time Domain Computation Framework 15 iii

7 2.1 VF-Based Macromodeling Frameworks and Supporting Techniques Data Algorithms Models Problems in the VF-Based Macromodeling Framework Macromodeling and Simulation in the Discrete-Time Domain Definitions of Discrete-Time Domain Models Poles in the Discrete-Time Domain Plane Conclusion Macromodeling Framework Advancements Via Discrete-Time Domain Computation Introduction Introduction of Vector Fitting Macromodeling in the Discrete-Time Domain Using Frequency-Sampled Response Data Pole Relocation of Real Poles Pole Relocation of Complex Poles Building the Macromodel Multi-port System Macromodeling in VFz Issues with Implementation Numerical Examples Macromodeling in the Discrete-Time Domain Using Time-Sampled Response Data Introduction to Time-Domain Macromodeling Introduction to Discrete Time-Domain Vector Fitting Pole Relocation of Real Poles iv

8 3.4.4 Pole Relocation of Complex Poles Building the Macromodel Convergence Analysis and Model Order Selection Numerical Examples Macromodeling in the Discrete-Time Domain using Hybrid-Domain Response Data Numerical Examples Macromodeling of Complex-Domain Responses Numerical Examples Macromodeling Using a P -norm Approximation Criterion Numerical Examples Conclusion VISA: Versatile Impulse Structure Approximation for Time-Domain Linear Macromodeling Introduction Formulation of VISA Numerator Calculation Denominator Calculation Convergence Analysis and Discussion VISA: Reformulation of Steiglitz-McBride Iteration Model Order Selection Comparison with Other Algorithms P -norm Approximation Criterion Multi-port System Macromodeling in VISA Comparison with the VFz family Numerical Examples v

9 4.6.1 Macromodeling of a Three-port RF Circulator Algorithm Performance Analysis P -norm Approximation Macromodeling of a Differential High-speed Backplane Channel Macromodeling of On-chip Passive Structures Conclusion Frequency Warping Technique for Universal Macromodeling Processes Introduction Frequency Warping via Bilinear Transform Numerical Examples Frequency Warping in The Time-Domain Macromodeling Frequency Warping in The Frequency-Domain Macromodeling Selection of the Warping Parameter Conclusion Conclusions and Future Work 99 7 Appendix - Macromodeling of Practical Testbenches Using VFz Introduction Numerical Examples Port (1,1), Port (2,1) and Port (4,1) of a Four-port Backplane Port (16,1) and Port (16,3) of a Sixteen-port Backplane Port (3,1) and Port (4,1) of a Four-port Backplane A Power-plane Pair of a Communication Board Three-port RF Circulator Four-port Transmission Line Conclusion vi

10 Bibliography 111 Publications and Awards 124 Vita 129 vii

11 List of Tables 3.1 Approximation result of the channel macromodeling using TD-VFz, VFz and HD-VFz: (a) number of frequency-sampled points, (b) number of timesampled points, (c) L 2 error norm in frequency responses, (d) L error norm in frequency responses, (e) L 2 error norm in time-domain responses, (f) L error norm in time-domain responses, and (g) CPU time (seconds) Fitting errors for the complex s-domain response modeling Fitting results for the complex z-domain response modeling Summary of contributions for the VF development Comparison between VISA and TD-VF in the circulator macrmodeling Comparison between different P -norm approximations in VISA, where 2 represents L -constrained 2-norm approximation Comparison between VISA and TD-VF in the benchmark examples. The bracketed value is the TD-VF result L 2 errors in z-domain macromodeling with two responses and two macromodeling algorithms, with comparison to z-domain macromodeling Implementation summary of different practical macromodeling examples viii

12 List of Figures 1.1 On-board interconnect structures On-chip interconnect structures [1]: (a) meander resistor, (b) insulatormetal-insulator capacitor, and (c) spiral inductor Signal integrity analysis through an eye diagram Common macromodeling flow in signal integrity analysis Response approximation in the macromodeling: (a) a frequency-sampled response and (b) a time-sampled response Equivalent circuit realization of a P in -input-ports and P out -output-ports macromodel Location of poles (+, and O): (a) in the s-domain plane and (b) equivalent pole location in the z-domain plane Typical pole locations for a six-pole z-domain macromodel of general lowpass (LP), bandpass (BP), and highpass (HP) responses. All poles are within a circle with radius = Geometry structure of the high-speed backplane Frequency responses of the channel example using VFz, compared to VF: (a) magnitude responses and (b) error in magnitude responses Converged pole locations of macromodel in the z-domain plane ix

13 3.4 Computation details with different number of iterations: (a) relative error and (b) condition number of the over-determined system equation matrix Magnitude responses of the channel macromodeling using VFz with noisecorrupted signal Relative error of the channel macromodeling with different response SNRs Computation details with different model orders: (a) relative error and (b) CPU time Magnitude responses of the inductor (SP-SMALL) macromodeling Illustrations of some CODESTAR testbench on-chip structures [1]: (a) meander resistor (RPOLY2-ME), (b) insulator-metal-insulator capacitor (CMIM), and (c) spiral inductor (SP-SMALL) Signal energy distribution of the power distribution network model Magnitude responses of the power distribution network macromodeling: (a) Port (6,10), (b) Port (7,13), and (c) Port (10,13) Relative error energy distribution of the power distribution network macromodel Frequency responses of the channel example using TD-VFz: (a) magnitude responses and (b) phase responses Time-domain transient simulation with the channel macromodel: (a) a random high-frequency digital input signal, and (b) output signal of the channel transmission Magnitude responses of the channel example using TD-VFz with noisecorrupted signal Hankel singular values (HSVs) of the system in the channel example x

14 3.17 Time-domain responses of the channel example using TD-VFz: (a) an exponential pulse input signal and (b) its output signal; (c) a step input signal and (d) its output signal; (e) a rectangular pulse input signal and (f) its output signal; and (g) a triangular pulse input signal and (h) its output signal Frequency responses of the channel example using HD-VFz, compared with VFz and TD-VFz: (a) magnitude responses and (b) error in the magnitude responses Time-domain responses of the channel example using HD-VFz, compared with VFz and TD-VFz: (a) magnitude responses and (b) error in the magnitude responses Initial pole placement and final pole locations in the complex s-domain response modeling Frequency responses of the modeling of the s-domain complex response, compared to conformal mapping method (Martin): (a) magnitude in the entire band, (b) magnitude in the passband and (c) group delay in the passband Frequency responses of the complex z-domain response modeling: (a) magnitude responses and (b) group delays Magnitude responses of the power distribution network: (a) approximation using L 2 norm, and (b) approximation using L norm Fitting in VISA: (a) input and output responses, (b) normalized impulse response and (c) FIR representation of the discretized response Illustration about the digital filtering relationship in the interpolation problem eq. (4.4) Illustration about the complementary signal of a triangular signal xi

15 4.4 Magnitude responses of S(u,v) of the circulator example using VISA in normalized frequency domain. S(u,v) is the scattering parameter at input port u and output port v Time responses of S(u,v) of the circulator example using VISA L 2 error in macromodeling using TD-VF and VISA in the circulator example: (a) 25th-order macromodel and (b) 32nd-order macromodel Hankel singular values (HSVs) of the system in the circulator example: (a) entire HSVs and (b) comparison with the macromodel error Time response of the differential channel example using VISA Magnitude response of the differential channel example using VISA in normalized frequency domain Relationship of the sampled location between the z-domain (original frequency) and the z-domain (warped frequency) with different warping parameters (γ) Responses of the backplane modeling using frequency warping. The differential channel with γ = 0.55: (a) frequency responses, and (b) its approximation error. The crosstalk between two channels with γ = 0.45: (c) frequency responses, and (d) its approximation error. Here (z) and (z warp ) represent the z-domain and z-domain, respectively L 2 error in the differential channel macromodeling using frequency warping: (a) with 0 γ 0.65 and (b) with 0.25 γ Maximum condition number during iterative calculations in two macromodeling examples using frequency warping with 0 γ L 2 error in the crosstalk macromodeling using frequency warping with 0 γ xii

16 xiii 5.6 L 2 error in the multi-port response macromodeling using frequency warping: (a) with 0 γ 0.95 and (b) with 0.15 γ Magnitude responses of the four-port backplane macromodeling: (a) Port (1,1), (b) Port (2,1) and (c) Port (4,1) Magnitude responses of the sixteen-port backplane macromodeling: (a) Port (16,1) and (b) Port (16,3) Magnitude responses of the four-port Intel backplane macromodeling: (a) Port (3,1) and (b) Port (4,1) Magnitude responses of the power-plane pair macromodeling Magnitude responses of the RF circulator macromodeling Magnitude responses of the four-port transmission line macromodeling: (a) Port (1,1) and (b) Port (1,4) Relative error energy distribution of the four-port transmission line model.. 109

17 List of Abbreviations s-domain DSP VFz z-domain TD-VFz EM HD-VFz HSV FIR IIR IEEE IC I/O LS MIMO RF SISO SK SI SNR continuous-time domain digital signal processing Discrete-Domain Vector Fitting discrete-time domain Discrete Time-Domain Vector Fitting electromagnetic Discrete-Time Hybrid-Domain Vector Fitting Hankel singular value finite-impulse-response infinite-impulse-response institute of electrical and electronics engineers integrated circuit input-output least-squares multiple-input-multiple-output radio frequency single-input-single-output Sanathanan-Koerner signal integrity signal-to-noise ratio xiv

18 xv SM VF VNA VISA VLSI Steiglitz-McBride Vector Fitting vector network analyzer Versatile Impulse Structure Approximation very-large-scale integration

19 Chapter 1 Introduction 1.1 IC design: In the Signal-Integrity-Aware Era Electronic systems, such as mobile communication systems [2] and computers [3], have become essential to our daily lives. From a system perspective, these electronic systems contain modules of integrated circuits (ICs), e.g., memory, datapath, control and inputoutput circuitry. Modules are connected by interconnects and power networks, including on-board/off-chip components (e.g., sockets, power/ground planes, via holes, wires, connectors and chip packages) and on-chip components (e.g., on-chip meander resistors / metal-insulator-metal capacitors / spiral inductors), as shown in Fig. 1.1 and Fig In low-speed circuit operation, interconnect networks do not distort transmitted signals much, and can be treated as ideal wire or generally modeled by equivalent resistor-capacitor circuits [4, 5]. However, as stated in the International Technology Roadmap for Semiconductors [6], design of IC chips is moving toward nano-scale. With the increasing operation frequency and decreasing feature size of ICs, high-frequency effects, such as signal delay, crosstalk and simultaneous switching noise, have become dominant factors that limit system performance [7, 8]. IC techniques are advancing so rapidly that existing design and verification 1

20 2 Sockets Wires between modules CPU chip packages Backplane connectors Figure 1.1: On-board interconnect structures. knowledge becomes obsolete. Signal integrity (SI) verification and signal-integrity-aware design have become popular practices in the IC design process to address such limitations, so as to ensure consistent signal transmissions and reliable power distributions in highspeed electronic systems [9 11]. In the post tape-out phase, SI analysis can be done by checking the eye diagrams of transmission channels on a circuit board obtained from measurement, as shown in Fig Meanwhile, in the pre tape-out phase, detailed modeling is required to capture the highfrequency behaviors of systems for simulation. However, a full-wave electromagnetic (EM) analysis over the global system is impractical. Reduced models with similar properties to the original system are therefore required. For simple transmission lines, transmission line theory [12] can be used to construct linear models [13, 14]. For structures with complicated geometry, such as packages, vias and radio-frequency (RF) objects, data-driven linear macromodeling is usually needed. In this thesis, data-driven linear macromodeling

21 3 Figure 1.2: On-chip interconnect structures [1]: (a) meander resistor, (b) insulator-metalinsulator capacitor, and (c) spiral inductor. (essentially a curve-fitting approach through the data) will be discussed in detail. 1.2 Signal Integrity: The Physical Interconnect Affection of the Electrical Performance in Circuits SI generally refers to problems that occur in the high speed circuit system due to physical interconnects. It happens as an EM phenomenon in nature [15]. However, as ICs moves toward the era of nano-scale and GHz-operation, SI problems now become the bottleneck of the high speed system design. SI issues disturb the signal voltage, and make the transmitted signal become ambiguous. As a result, the quality of the signal transmission and the circuit performance are degraded by the disturbance, especially in high speed circuits. Discussions about SI can be found in [4, 5], and some important issues will be highlighted in this subsection. In high speed systems, there are some common SI problems: Reflections: It is caused by the impedance discontinuities along the signal transmission path. The mismatched impedance causes signal reflection, overshoot, undershoot and ringing during the transmission. This issue becomes apparent when there

22 4 Figure 1.3: Signal integrity analysis through an eye diagram. is a high impedance open or a low impedance at the output driver. Crosstalk: It is caused by the significant EM couplings between multiple parallel transmission lines. Signals from adjacent lines are coupled over and become disturbances in other lines. This issue becomes apparent when transmission lines are uniformly parallelized. Simultaneous switching noise: It is caused by the significant parasitics of the power/ground delivery system. The driver switching transient and parasitics cause the voltage fluctuation between power and ground plane, and the fluctuated power supply disturbs signals on the signal plane. This issue becomes apparent when the circuit has a high switching frequency, a low operating voltage and is connected with many I/Os. Generally, SI issues can be classified into three levels: chip level, package level and board level. Each levels has their specific physical characteristics, modeling considerations, modeling procedures and SI solutions. In particular, chip level SI is different from package/board level SI based on the following reasons:

23 5 1. Board level interconnects have a higher ratio between interconnect transmission time and bit period, that is, board level interconnects are considered as electrically long for a fast switching transmission. As a result, received signals always contain propagation delays. On the other hand, in the chip level, the echoes of previous pulses have not attenuated due to in-sufficient rebounding in the interconnect structure and disturb the main pulse. 2. Board level interconnects carry more current, and induce a magnetic/inductive crosstalk instead of a capacitive crosstalk in chip level interconnects. 3. Board level interconnects has a bigger cross-sectional size and a smaller series resistance. However, the high frequency component of the signal is attenuated by the skin effect in the metal of the interconnect and the dielectric loss tangent of the circuit board. Based on these classifications, each level has their specific SI concerns: Chip level: Improper I/O buffer design and inadequate return current paths; Package Level: High package inductance, mismatched traces, improper routing and placement, insufficient protection for critical paths, and inadequate return current paths; Board Level: Crosstalk, reflections, signal attenuation and EM interference from the adjacent boards. Because of the development of emerging interconnect technologies [8], new issues, new considerations and new necessities have been raised for developments of the SI analysis process: 1. Due to the progressive development of circuit system integration (e.g., system-inpackage (SIP) and system-on-chip (SOC)), circuit feature minimization and circuit switching acceleration, broad level SI problems will appear in the chip level soon.

24 6 Macromodeling Response Characterization Data pre-processing f(x) + Simulation Model post-processing Figure 1.4: Common macromodeling flow in signal integrity analysis. 2. Because of the growing popularity of 3D circuit system integration, the future interconnect systems will have more vertical connections (vias) and a more complicated geometry. Therefore, the interconnect structures should be fully modeled through the macromodeling process. 3. The mixed integration of noise sensitive components (e.g. radio frequency and sensor chips) with noisy digital chips on the same board/chip will bring out more high frequency EM interference problems in circuit systems. Furthermore, thermal problems in advanced interconnect structures affect physical properties and performances of interconnect systems. Detailed modeling is required to capture these newly raised behaviors. 1.3 Macromodeling: System Identification Problem in Signal/Power Integrity Macromodeling is a standard part of the SI analysis process. Through macromodeling, the interconnect network can be modeled as a macromodel for an efficient simulation. A

25 7 common data-driven macromodeling flow is shown in Fig The sampled structure responses can be obtained by exciting one input port at a time and then computing or measuring the responses at (some or all of) the output ports (Response Characterization). By approximating the sampled frequency-dependent or time-dependent system response data, a macromodel is generated to replace the original large-order system by a smaller-order one with similar input-output (I/O) behaviors (Macromodeling). The macromodel is used to produce spectra and waveforms for SI analysis and/or coupled with other circuit model blocks (e.g., logic devices) for global simulation (Simulation). Moreover, supporting preand post-processing techniques are available to modify the macromodel characteristics and enhance the simulation performance. To model a complicated geometry structure, the I/O characteristics of the structure are described by the I/O response data, either calculated or measured. Generally, for a singleport system, macromodeling techniques intend to fit a linear time-invariant (LTI) system to the desired continuous-time frequency domain (s-domain) response H (s) at a set of calculated/sampled points at the I/O ports. The model is usually a state-space system or a rational transfer function with a set of predefined basis {φ n } (numerator-denominator form) H (s) N (s) D (s) = N b n φ n (s), (1.1) N bn φ n (s) where b n,b n R and N is the macromodel order. H (s) can also be approximated using a summation of partial fraction basis and a unity basis with their model parameters {c n } and d (pole-residue form), n=1 n=1 ( H (s) N (s) N D (s) = n=1 c n s + α n ) + d. (1.2)

26 8 Magnitude Magnitude Figure 1.5: Response approximation in the macromodeling: (a) a frequency-sampled response and (b) a time-sampled response. Usually, the algorithm is required to fit tens of ports in the original system, where each port contains hundreds of frequency-sampled data points. An illustration of the response approximation is shown in Fig Therefore, the linear structure macromodeling can be regarded as a large-scale broadband system identification problem. Comprehensive studies about system identification can be found in [16 18]. In macromodeling, the system identification process must meet several specific and stringent constraints, namely, 1. Accuracy and physical consistency in response approximation; 2. Low computation complexity; 3. Numerically robust computation; 4. Minimal manual intervention during calculation. In the L 2 -norm sense with N s sampled points, the optimal model of a system can be obtained through minimizing the following objective function N s min N (s k ) D (s k ) H (s k). (1.3) 2 k=1

27 9 However, this is a numerically sensitive non-linear problem with no prior information about the exact pole and zero locations of the system. The response is usually approximated using Prony s method [19] for a coarse solution or other identification frameworks for a finer solution. Such frameworks include s-domain Sanathanan-Koerner (SK) iteration [20] or equivalent discrete-time domain (z-domain) Steiglitz-McBride (SM) iteration [21]. The objective function of the SK iteration in the ith iteration is N s min N (i) (s k ) D (i 1) (s k ) D(i) (s k ) D (i 1) (s k ) H (s k). (1.4) 2 k=1 By arranging the weighting function σ (i) (s) := D (i) (s) / D (i 1) (s), the model parameters can be determined via a least-squares (LS) problem N (i) (s) D (i) (s) D (i) (s) D (i 1) (s) }{{} (σh) (i) (s) D(i) (s) D (i 1) (s) } {{ } σ (i) (s) H (s) 0. (1.5) If a monomial power series basis function is used in eq. (1.5) for broadband macromodeling, i.e., φ n (s) = s n, the traditional SK iteration approach will suffer from an ill-conditioned Vandermonde matrix calculation [22], and thereby will not satisfy the macromodeling requirements from a numerical perspective. 1.4 Digital Signal Processing: Processing Sampled and Discretized Signals Digital signal processing (DSP) represents, transforms and manipulates digitized (discretized) signals and information in an organized and meaningful manner [23]. Signal processing starts from the classical numerical analysis and analog systems analysis, the process was then digitalized in the 1950s and used for digital control systems. DSP has

28 10 been widely used in different applications, such as communications [24] and multimedia [25]. For example, a communication system performs modulation, compression, and recognition of signals using DSP techniques. In most applications, signals are mainly continuous-time signals. To perform DSP, response is sampled as discrete-time signal using an analog-to-digital converter, and transformations (e.g., bilinear transformation) are used for the analytical study. Besides performing temporal/spatial processing, signals can be processed and analysed in the frequency domain (e.g., filtering, transformation, identification) by efficient Discrete Fourier Transform (DFT), or other domains (e.g., autocorrelation domain and wavelet domain). Compared to its analog counterpaart, DSP has a more numerically robust computation. As signals are represented as a series of discrete numbers in DSP, error detection, error correction and signal compression can be done through algorithms. In addition, the processing computations, such as convolution and transformation, can be efficiently handled through digital processors, such as application-specific integrated circuits (ASICs), field-programmable gate arrays (FPGAs) or specialized digital signal processing processors (DSP processors). 1.5 Research Motivations and Dissertation Outline The macromodeling process has substantial progress in the past few years. Active and intensive work have been carried out in this area in both the academia and the industry for the following reasons: 1. When ICs move toward the era of nano-scale and 3D integration, more SI issues will be raised, highly detailed simulations will then become necessary, and computations will become large scale and more complicated. Progressive macromodeling techniques and SI analysis techniques are required to analyse those emerging interconnect systems.

29 11 2. Most SI simulation problems are large scale and requires long computation time, which costs critical time delays in time-to market. To keep pace with the short design process requirement, novel techniques are required to make SI analysis practical with a reasonable amount of computational power. 3. Macromodeling is a highly non-linear computation process with different practical requirements in different modeling situations, there is no optimal algorithm in terms of computational complexity and approximation optimality. 4. Besides improving the macromodeling algorithm from mathematical (mostly controltheoretic) perspective, pre- and post-processing techniques and improvements from non-control-theoretic perspective have been less explored. With sampled response is a sampled and discretized signal sequence as the fundamental concept, this thesis explores the feasibility and benefits of applying DSP techniques to the macromodeling process, focusing on 1. Improving the functionality and automation of the approximation process; 2. Increasing the fitting accuracy of the approximation process; 3. Reducing the computation time of the macromodeling procedure. In this doctoral study, we develop new methodologies, new generalizations of existing methodologies and new pre-/post-processing techniques to achieve the research outcomes. We compare their performances with existing methodologies using industrial benchmark examples. The thesis is structured as follows:

30 12 Chapter 2 - Macromodeling Process Development in the Vector Fitting Framework and Discrete-Time Domain Computation Framework To build up a substantial foundation for the future development of the macromodeling process, we first explore the macromodeling process in signal/power integrity analyses in a system identification perspective. We study the major developments of macromodeling process, especially developments based on the widely adopted Vector Fitting (VF) framework, along three directions: data, algorithms and models. Furthermore, since research committees have recently raised their interest in the discrete-time domain (z-domain) macromodeling development [26], we also discuss some features and advantages of z-domain computation in the macromodeling process. Chapter 3 - Macromodeling Framework Advancements Via Discrete Time Domain Computation To enhance the functionality of the widely adopted, iterative based Vector Fitting framework, we propose a family of improvements based on the root of the z-domain computation. First, we propose the z-domain counterpart of VF (VFz) [27], which uses z-domain partial fraction basis to seek a rational approximation to the desired response. This improves the numerical conditioning and convergence in broadband frequency-sampled system macromodeling. Second, we extend VFz to its discrete time-domain (TD-VFz) [28] variants for the specific time domain macromodeling purpose. TD-VFz does not require discretizing the continuous-time convolution integrals for each iteration and it provides a priori model order selection and a time delay extraction as pre-processing steps. Third, we also propose the discrete-time hybrid-domain (HD-VFz) variants [29], which model responses with a better hybrid-domain accuracy through providing extra informative data. Fourth, in order to improve the functionality of VF, we also extend VF to the effective asymmetrical response modeling of complex-domain systems (e.g., complex filter) by relaxing the

31 13 complex conjugate pole pair restriction in VF [30] and VFz [31]. Fifth, we develop a versatile macromodeling adoption through a P -norm approximation expansion [32]. At last, we model various practical examples via VFz to demonstrate the working process and its excellent performance, as shown in the appendix. Chapter 4 - VISA: Versatile Impulse Structure Approximation for Time-Domain Linear Macromodeling To avoid numerically sensitive initial guess and expensive computation in time-domain macromodeling process, we develop a rational function macromodeling algorithm called Versatile Impulse Structure Approximation (VISA) [33, 34]. The idea is to regard the system response as the impulse response of a finite-impulse-response (FIR) filter, and then apply infinite-impulse-response (IIR) filter approximation techniques to generate the macromodel. By applying the idea of Walsh theorem and complementary signal, this approach can be interpreted as a non-pole-based Steiglitz-McBride (SM) iteration without initial guess assignment and eigenvalue computation, which is numerically simple (i.e., robust and efficient) to determine macromodel parameters. Chapter 5 - Frequency Warping Technique for Universal Macromodeling Processes To alleviate ill-conditioned computation problem in the linear-structure macromodeling process, we propose frequency warping, a spectral pre-processing scheme, by changing the response sampling distribution. Frequency warping transforms the structure response (time-/frequency-sampled data), in order to introduce a numerically favorable fitting in the frequency domain, and to improve the fitting accuracy during the macromodeling process. We also suggest that the frequency warping can be applied to multi-port configurations, different structure responses and algorithms through a simple bilinear transform.

32 14 Chapter 6 - Conclusions and Future Work Based on the research study, we draw some conclusions and propose a few research topics for further investigations.

33 Chapter 2 Macromodeling Process Developments in the Vector Fitting Based Framework and Discrete-Time Domain Computation Framework While transient simulation and circuit model construction can be dated back to 1950 s [35], some macromodeling approaches have been developed only very recently, e.g., curve fitting with pole-clustering [36], rational function approximation [37, 38], subband filter cascading [7, 39], convex programming [40] and vector fitting (VF) [41]. In particular, VF is regarded as a robust and simple broadband macromodeling technique [10, 42, 43], and widely adopted in the SI community. In this chapter, we discuss the development of the macromodeling process, based on the VF framework. We also discuss the development of the discrete-time domain (z-domain) macromodeling, which is the core idea of this thesis. 15

34 VF-Based Macromodeling Frameworks and Supporting Techniques Recognizing its simple implementation, good performance and versatile extensibility, VF framework has been widely applied to the macromodeling of linear structures in SI analyses. It has been widely used in other areas [10, 27, 29, 30, 41, 44 46]. A number of generalizations and extensions of VF have been proposed to improve the macromodeling performance and thus better align with various identification requirements [22, 47 62]. In this section, we will discuss the development of macromodeling along three directions: data, algorithms and models Data Data describe the system response, and are generally obtained by measurements (e.g., vector network analyzer (VNA)) or EM simulations (e.g., Nexxim [63]). Since data content can affect the properties and quality of the resultant macromodel, different considerations and techniques have been proposed to ensure the input data are maximally informative for the identification purpose. Input Data Choices Continuous-time frequency-sampled response data H (s) are often used for macromodeling, which capture the high-frequency behaviors of the system [41]. Frequency-sampled data examples include scattering parameters (S-parameters) for RF objects and admittance parameters (Y -parameters) for interconnects. Other data choices, such as time-sampled response data (input and output response X [n] and Y [n]) [50], frequency response derivative H (s) [51], phase response H (s) [52] and magnitude response H (s) [53], can also be used for other different identification purposes.

35 17 Pre-Processing of Data The system response should correctly describe the system. However, some problems, such as data burst, defects, missing and noise disturbance, may occur during the process of data collection. Some information may be lost, and difficulties and failures in approximation may arise. To generate a correct macromodel, the data must be meaningful (e.g., passive and causal, as explained later). To ensure that, data pre-processing, such as causality and passivity verifications of input data and delay extraction by (generalized) Hilbert transform [64], is required. Furthermore, causality-constrained data interpolation is developed to generate consistent direct current (DC) and low-frequency data, which is necessary for simulation but usually not provided in the frequency-sampled data [64]. In addition, a large data set or broadband responses usually have a large variance and may result in ill-conditioned calculation. Pre-filtering techniques, in this scenario, can be used to change the distribution of noise and bias, so as to give a better fitting of the significant frequency range and a numerically favorable calculation with small computational costs. An appropriate adaptive [65] or deterministic data selection process and response weighting can also help improve the approximation. Pre-Processing of Model A priori configuration of macromodels should be chosen according to the knowledge of the algorithms (such as SK iteration) and data for a convenient approximation. For example, a priori model order selection, by applying experimental observation of the frequency response to frequency-sampled data [54] and time-sampled data [61], helps generate a minimal-size macromodel for efficient simulations with accuracy control.

36 Algorithms Given a set of input data, an algorithm is used to determine the model parameters. A good algorithm should have an appropriate identification criterion and should be simple and robust in terms of numerical implementation. We will first discuss the algebraical minimization criteria, and then the numerical implementation for a numerically favorable model parameter calculation. Identification Criterion and Framework The selection of approximation criteria is important for model approximation. The approximated model should be reliable, obtained within a reasonable computation time, and should admit an exact description of the real system. SK iteration with an L 2 -norm error prediction is usually used since it is applicable to different models and situations. Other criterion extensions have been recently developed for specific applications. Massive-port macromodeling: Multi-port macromodeling can be handled by stacking the system equation matrices of responses of all the ports into a single column of over-determined equation. However, modeling systems with a large number of ports (e.g., package parasitic networks and electromagnetic-aware circuits) is challenging in terms of complexity and robustness of computation. To model a system with an arbitrary number of ports, an alternative is proposed to approximate the eigenpairs rather than the matrix elements [56], which gives a more accurate approximation for systems with a large ratio between the largest and smallest eigenvalues. Parametric macromodeling: Nano-scale high-frequency circuit simulation and design must consider variabilities in geometry and material properties generated during the manufacturing process. In order to accurately model the behavior and reduce the computation time of repeated simulations, a parametric macromodel is used to describe the variational

37 19 structures, for example, H (s,g) ( Ns P n=0 Ns n=0 ) p=1 b npϕ p (g) φ n (s) ), (2.1) φ n (s) ( P p=1 b np ϕ p (g) where φ n (s) is the frequency-dependent basis and ϕ p (g) is the variability-dependent basis with a single variational parameter g and P samples in the variability domain. The variational structures can then be described by a macromodel with a polynomial basis or rational function basis [43, 59, 60]. Numerical Implementation Due to the nature of iterative calculation, its implementation is usually numerically sensitive. Although VF solves the ill-conditioned high-frequency system identification by a partial-fraction basis, some other problems, such as inappropriate initial guess and noisecontaminated responses, tend to damage the convergence of the algorithm. Some improvements have been proposed to address the convergence problem. Initial poles and applied basis: The algorithm generates a set of model parameters (b n and b n in eq. (1.1)) according to the given set of basis φ (s), the sampled data H (s) and the initial poles {α n }. The selection of the basis will affect the conditioning of the system equation matrix in eq. (1.5) and the solution accuracy. One selected approach is to choose an appropriate set of initial poles by a simple calculation (e.g., Prony method [19]), or intuitively assigned by a set of weakly-damped initial poles [41]. Another approach is to select a robust basis for calculation to minimize the numerical disturbance due to the inappropriate set of poles. For instance, the orthonormal basis φ or n (s) [48] has been proposed, namely, φ or n (s) = κ n 2R (αn ) ( n 1 j=1 s α j s + α j ) 1 s + α n, (2.2)

38 20 where κ n is the normalization coefficient and denotes complex conjugate. Orthonormal basis, from a mathematical perspective, reduces the condition number of the system equation matrix. Details about orthonormal basis are discussed in [66]. Different situations, such as modeling responses with repeated poles [48], call for different basis generalizations. Macromodeling with noisy signals: Experiences show that the algorithm convergence is slowed down in noise-contaminated signals and is biased in the low-frequency region, as the LS normalization of equation solving is impaired. To address this problem, a relaxed LS normalization is adopted [47, 55]. The relaxation improves the normalization of the transfer function coefficients and the linearization of the SK iteration without affecting the convergence Models The macromodel (model) describes the I/O characteristics of the approximated system, for SI analysis and coupled simulation with other circuit models. The model should be accurate, physically consistent and of low complexity for simulation. Necessary post-processing techniques are adopted to ensure a correct simulation. Post-Processing for a Physically Consistent Model The macromodel should be physically consistent, i.e., real-valued, stable, passive and causal [67]. Some important issues will be highlighted here. Real-valued: Real-valued macromodels do not generate complex-valued responses for real-valued input data. This means the poles in macromodels must be either real poles or complex conjugate pairs. If complex poles are not restricted to conjugate pairs, numerical computations (e.g., VF) may produce numerical errors and generate complex-valued macromodels. To construct a real-valued macromodel, transformations of the numerical

39 21 implementation are required [41]. Furthermore, complex-valued computations are separated into real and imaginary parts to avoid numerical errors, at the expense of an increased problem size. Stable: Stable macromodels do not generate responses beyond limits for any bounded input signal. An unstable pole (i.e., one with real part > 0 in the s-domain) can be stabilized through a non-linear pole flipping. The flipping on the other hand may cause phase distortion. Passive: Passive macromodels do not generate energy, yet most macromodeling algorithms (e.g., VF) may generate slightly non-passive macromodels due to numerical errors. To ensure passivity, perturbation of model parameters is required to passify the model [68]. Causal: Causal macromodels do not generate output signals according to future input. However, purely rational function macromodeling of electrically-long structures (i.e., responses with signal delays) often suffer from inapplicable fitting, and generate non-causal models. A reformulated macromodeling approach is developed [58]: with D obtained time delays {τ d }, the response can be fitted via N D n=0 d=1 H (s) b ndφ n (s)e sτ d N. (2.3) n=0 b nd φ n (s) Post-Processing for Simulation The approximant macromodel is used to generate the frequency response, time-domain reflectometry (TDR) waveforms, time-domain transmissometry (TDT) waveforms and eye diagrams for channel analysis [69], or coupled with other models for overall simulation. For efficient simulation and analysis, macromodels need to be fully integrated with simulation tools. The macromodel can be described by a pole-residue form in Matlab Simulink or Verilog-A description for high-level simulation [70]. The macromodel can also be described as an equivalent circuit in a SPICE netlist and integrated into the non-linear circuit